CS285 Designing Viae Globi (Roads on a Sphere) Carlo H. Séquin University of California, Berkeley Inspired by Brent Collins Gower, Missouri

“Hyperbolic Hexagon” by B. Collins u 6 saddles in a ring u 6 holes passing through symmetry plane at ±45º u = “wound up” 6-story Scherk tower u Discussion: What if … l we added more stories ? l or introduced a twist before closing the ring ?

Closing the Loop straight or twisted

“Sculpture Generator I”, GUI

Base Geometry: One Scherk Story u Hyperbolic Slices ==> Triangle Strips u Pre-computed -- then warped into toroid

Brent Collins “Hyperbolic Hexagon II”

Hyper-Sculpture: Family of 12 Trefoils W=2 W=1 B=1 B=2 B=3 B=4

Brent Collins’ Pax Mundi 1997: wood, 30”diam. 2006: Commission from H&R Block, Kansas City to make a 70”diameter version in bronze. My task: to define the master geometry. CAD tools played important role.

How to Model Pax Mundi... u Already addressed that question in 1998: u Pax Mundi could not be done with Sculpture Generator I u Needed a more general program ! u Used the Berkeley SLIDE environment. u First: Needed to find the basic paradigm   

Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis- or base-ball;  2-period Gabo curve.

2-period “Gabo Curve” u Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used (symmetry!).

4-period “Gabo Curve” Same construction as for as for 2-period curve

Pax Mundi Revisited u Can be seen as: Amplitude modulated, 4-period Gabo curve

SLIDE SLIDE = Scene Language for Interactive Dynamic Environments Developed as a modular rendering pipeline for our introductory graphics course. Primary Author: Jordan Smith u Based on OpenGL and Tcl/tk. u Good combination of interactive 3D graphics and parameterizable procedural constructs.

SLIDE Example: Klein Bottle Final Project CS 184, Nerius Landys & Shad Roundy

SLIDE Example Bug’s Life Final Project CS 184, David Cheng and James Chow

SLIDE as a Design Tool u SLIDE originally a modular rendering tool. u Later enhanced to serve as a CAD tool: l Spline curves and surfaces l Morphing sweeps along such curves l 3D warping module (Sederberg, Rockwood) l Many types of subdivision surfaces u These are key elements for a 2 nd Generation Sculpture Generator

SLIDE-UI for Knot Generation

SLIDE-GUI for “Pax Mundi” Shapes Good combination of interactive 3D graphics and parameterizable procedural constructs.

2-period Gabo sculpture Tennis ball – or baseball – seam used as sweep curve.

Viae Globi Family (Roads on a Sphere) Viae Globi Family (Roads on a Sphere) periods

Via Globi 5 (Virtual Wood) Wilmin Martono

Modularity of Gabo Curve Generator u Sweep Curve Generator: l Gabo Curves as B-splines u Cross Section Fine Tuner: l Paramererized shapes u Sweep / Twist Controller

Sweep / Twist Control u How do we orient, move, morph... the cross section along the sweep path ? Natural orientation with Frenet frame Torsion Minimization: Azimuth: tangential / normal 900° of twist added.

Extension: Free-form Curve on a Sphere Spherical Spline Path Editor (Jane Yen) Nice smooth interpolating curves through sparse data points

Many Different Viae Globi Models

Maloja

Extending the Paradigm: Aurora-M u Simple path on sphere, u but more play with the swept cross section. u This is a Moebius band. u It is morphed from a concave shape at the bottom to a flat ribbon at the top of the flower.

Paradigm Extension: Sweep Path is no longer confined to a sphere! Music of the Spheres (Brent Collins)

Allows Knotted Sweep Paths Chinese Button Knot

Really Free-form 3D Space Curves Figure-8 knot

Using Another Cross Section Galapagos 6

Target Geometry Constraints: Bronze, 70” diameter Less than 1500 pounds Less than $50’000 Maintain beauty, strength Minimize master geometry

Emulation u Use smoother quintic spline u Make 4 identical parts (needs extra control point) u Add a few “warp” parameters

Emulation; Define Master Pattern u Use 4 copies. u Master to make a mold from. Alignment tab

Model of Master Part Made with FDM u 4 pieces make the whole sculpture

Joe Valasek’s CNC Milling Machine u Styrofoam milling machine

Design of Two-Part Master u Alignment tabs for easy assembly

Subdivide into Two Master Segments

Machined Master Pattern #2

(Cut) Master  Silicone Rubber Mold

Mold  Several (4) Wax Copies

Spruing the Wax Parts for Casting

Ceramic Slurry Shell Around Wax Part

Shell Ready for Casting

Casting with Liquid Bronze

The Freed Bronze Cast

Assembling the Segments

The “Growing” Ribbon

The Single Support Point

Grinding the Welded Seams, Polishing the Surface

Applying Patina

Ready for Shipping

The Move Around the Building (Jan.18)

Front Door H&R Block Building

Steve Tightening the Bolts

Brent Polishing Our Baby